Worksheet: Combining Functions

In this worksheet, we will practice adding, subtracting, multiplying, or dividing two given functions to create a new function and identifying the domain of the new function.

Q1:

What is the domain of the quotient ๐‘“ ๐‘” , in terms of the domains of ๐‘“ and ๐‘” ? Assume that both domains are subsets of the set of real numbers.

  • A the union of the domain of ๐‘“ and the domain of ๐‘”
  • B the intersection of the domain of ๐‘“ and the domain of ๐‘”
  • C the larger of the domain of ๐‘“ and the domain of ๐‘”
  • D the intersection of the domain of ๐‘“ and the domain of 1 ๐‘”
  • Ethe difference between the domain of ๐‘“ and the domain of ๐‘”

Q2:

Determine the common domain of the functions ๐‘› ( ๐‘ฅ ) = โˆ’ 7 ๐‘ฅ โˆ’ 7 ๏Šง and ๐‘› ( ๐‘ฅ ) = โˆ’ 8 ๐‘ฅ โˆ’ 6 4 ๏Šจ ๏Šจ .

  • A โ„ โˆ’ { โˆ’ 8 , โˆ’ 7 , 8 }
  • B โ„ โˆ’ { โˆ’ 8 , 8 }
  • C โ„ โˆ’ { 7 , 8 }
  • D โ„ โˆ’ { โˆ’ 8 , 7 , 8 }
  • E โ„ โˆ’ { โˆ’ 8 , โˆ’ 7 }

Q3:

If ๐‘“ and ๐‘” are two real functions where ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 1 ๐‘ฅ + 3 ๐‘ฅ โˆ’ 4 ๏Šจ and ๐‘” ( ๐‘ฅ ) = ๐‘ฅ + 3 , determine the value of ( ๐‘“ + ๐‘” ) ( โˆ’ 4 ) if possible.

  • A โˆ’ 6 5
  • B โˆ’ 1
  • C โˆ’ 6
  • Dundefined

Q4:

Find the domain of the function ๐‘“ ( ๐‘ฅ ) = โˆš ๐‘ฅ + 3 + โˆš ๐‘ฅ โˆ’ 7 ๏Žข .

  • A [ 3 , โˆž )
  • B ( โˆ’ 3 , โˆž )
  • C [ 7 , โˆž )
  • D [ โˆ’ 3 , โˆž )

Q5:

If ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ + 1 and ๐‘” ( ๐‘ฅ ) = ๐‘ฅ + 1 ๏Šจ , then find and fully simplify an expression for ( ๐‘“ โ‹… ๐‘” ) ( ๐‘ฅ ) .

  • A ๐‘ฅ + ๐‘ฅ + 1 ๏Šฉ ๏Šจ
  • B ๐‘ฅ + ๐‘ฅ + 2 ๏Šจ
  • C ๐‘ฅ + ๐‘ฅ + 1 ๏Šฉ
  • D ๐‘ฅ + ๐‘ฅ + ๐‘ฅ + 1 ๏Šฉ ๏Šจ

Q6:

If ๐‘“ โ„ โ†’ โ„ : where ๐‘“ ( ๐‘ฅ ) = 4 ๐‘ฅ โˆ’ 4 , and ๐‘” [ โˆ’ 8 , โˆ’ 2 ) โ†’ โ„ : where ๐‘” ( ๐‘ฅ ) = 5 ๐‘ฅ + 5 , find the value of ( ๐‘“ + ๐‘” ) ( 5 ) if possible.

  • A16
  • B46
  • C36
  • Dundefined

Q7:

Given that ๐‘“ โˆถ โ„ ๏Šฐ โ†’ โ„ , where ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 1 9 , and ๐‘” โˆถ [ โˆ’ 2 , 1 3 ] โ†’ โ„ , where ๐‘” ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 6 , evaluate ( ๐‘“ โ‹… ๐‘” ) ( 7 ) .

  • A โˆ’ 2 4 0
  • B724
  • C โˆ’ 7 7 4
  • D โˆ’ 1 2

Q8:

If ๐‘“ and ๐‘” are two real functions where ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ + 9 ๐‘ฅ + 1 5 ๐‘ฅ + 5 4 ๏Šจ and ๐‘” ( ๐‘ฅ ) = ๐‘ฅ + 8 , determine the value of ( ๐‘“ โˆ’ ๐‘” ) ( โˆ’ 6 ) if possible.

  • A โˆ’ 5 3
  • B โˆ’ 2
  • C1
  • Dundefined

Q9:

Given that , , and , determine in its simplest form.

  • A
  • B
  • C
  • D
  • E

Q10:

Given that ๐‘› ( ๐‘ฅ ) = 5 ๐‘ฅ โˆ’ 8 2 5 ๐‘ฅ โˆ’ 4 รท 2 5 ๐‘ฅ โˆ’ 3 0 ๐‘ฅ โˆ’ 1 6 1 2 5 ๐‘ฅ + 8 ๏Šง ๏Šจ ๏Šจ ๏Šฉ , ๐‘› ( ๐‘ฅ ) = 2 5 ๐‘ฅ โˆ’ 4 5 0 ๐‘ฅ โˆ’ 2 0 ๐‘ฅ + 8 ๏Šจ ๏Šจ ๏Šจ , and ๐‘› ( ๐‘ฅ ) = ๐‘› ( ๐‘ฅ ) ร— ๐‘› ( ๐‘ฅ ) ๏Šง ๏Šจ , simplify the function ๐‘› , and determine its domain.

  • A ๐‘› = 2 , domain = โ„ โˆ’ ๏ฌ โˆ’ 2 5 , 2 5 , 8 5 ๏ธ
  • B ๐‘› = 1 2 , domain = โ„ โˆ’ ๏ฌ โˆ’ 2 5 , 2 5 ๏ธ
  • C ๐‘› = 2 , domain = โ„ โˆ’ ๏ฌ โˆ’ 2 5 , 2 5 ๏ธ
  • D ๐‘› = 1 2 , domain = โ„ โˆ’ ๏ฌ โˆ’ 2 5 , 2 5 , 8 5 ๏ธ
  • E ๐‘› = 5 2 , domain = โ„ โˆ’ ๏ฌ โˆ’ 2 5 , 2 5 , 8 5 ๏ธ

Q11:

Given that ๐‘“ โˆถ ( โˆ’ โˆž , 4 ) โ†’ โ„ ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ + 5 ๏Šง ๏Šง s u c h t h a t and ๐‘“ โˆถ ( โˆ’ 8 , 6 ) โ†’ โ„ ๐‘“ ( ๐‘ฅ ) = 2 ๐‘ฅ + 1 3 ๐‘ฅ + 1 5 , ๏Šจ ๏Šจ ๏Šจ s u c h t h a t find ๏€ฝ ๐‘“ ๐‘“ ๏‰ ( ๐‘ฅ ) ๏Šจ ๏Šง and state its domain.

  • A 2 ๐‘ฅ + 3 , ๐‘ฅ โˆˆ ( โˆ’ 8 , 6 )
  • B 2 ๐‘ฅ + 3 , ๐‘ฅ โˆˆ ( โˆ’ โˆž , 4 )
  • C 2 ๐‘ฅ + 3 , ๐‘ฅ โˆˆ ( โˆ’ 8 , 4 )
  • D 2 ๐‘ฅ + 3 , ๐‘ฅ โˆˆ ( โˆ’ 8 , 4 ) โˆ’ { โˆ’ 5 }
  • E 2 ๐‘ฅ + 3 , ๐‘ฅ โˆˆ ( โˆ’ โˆž , 4 ) โˆ’ { โˆ’ 5 }

Q12:

Given that ๐‘“ โˆถ โ„ โ†’ โ„ ๐‘“ ( ๐‘ฅ ) = โˆ’ 3 ๐‘ฅ โˆ’ 4 w h e r e and ๐‘” โˆถ ( 1 , 7 ) โ†’ โ„ ๐‘” ( ๐‘ฅ ) = โˆ’ 2 ๐‘ฅ โˆ’ 4 , w h e r e find the value of ๏€ฝ ๐‘“ ๐‘” ๏‰ ( โˆ’ 1 ) if possible.

  • A โˆ’ 1
  • B 1 2
  • C0
  • Dnot defined

Q13:

Given that ๐‘“ โˆถ ( โˆ’ โˆž , 2 ] โŸถ โ„ ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ + 5 ๏Šง ๏Šง s u c h t h a t and ๐‘“ โˆถ ( โˆ’ โˆž , โˆ’ 1 ) โŸถ โ„ ๐‘“ ( ๐‘ฅ ) = 2 ๐‘ฅ โˆ’ ๐‘ฅ โˆ’ 6 , ๏Šจ ๏Šจ ๏Šจ s u c h t h a t find ๏€ฝ ๐‘“ ๐‘“ ๏‰ ( ๐‘ฅ ) ๏Šจ ๏Šง and state its domain.

  • A 2 ๐‘ฅ โˆ’ ๐‘ฅ โˆ’ 6 ๐‘ฅ + 5 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ โˆž , โˆ’ 1 ]
  • B 2 ๐‘ฅ โˆ’ ๐‘ฅ โˆ’ 6 ๐‘ฅ + 5 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ โˆž , โˆ’ 1 )
  • C 2 ๐‘ฅ โˆ’ ๐‘ฅ โˆ’ 6 ๐‘ฅ + 5 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ โˆž , 2 ]
  • D 2 ๐‘ฅ โˆ’ ๐‘ฅ โˆ’ 6 ๐‘ฅ + 5 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ โˆž , โˆ’ 1 ) โˆ’ { โˆ’ 5 }
  • E ๐‘ฅ + 5 2 ๐‘ฅ โˆ’ ๐‘ฅ โˆ’ 6 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ โˆž , โˆ’ 1 ) โˆ’ { โˆ’ 5 }

Q14:

If ๐‘“ and ๐‘” are two real functions where ๐‘“ ( ๐‘ฅ ) = ๏ญ 2 ๐‘ฅ + 2 ๐‘ฅ < โˆ’ 3 , ๐‘ฅ โˆ’ 4 โˆ’ 3 โ‰ค ๐‘ฅ < 0 , i f i f and ๐‘” ( ๐‘ฅ ) = 5 ๐‘ฅ determine the domain of the function ๏€ฝ ๐‘” ๐‘“ ๏‰ .

  • A [ โˆ’ 3 , 0 )
  • B ( โˆ’ โˆž , โˆ’ 3 )
  • C โ„ โˆ’ { 0 }
  • D ( โˆ’ โˆž , 0 )

Q15:

If ๐‘“ and ๐‘” are two real functions where ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 5 ๐‘ฅ ๏Šจ and ๐‘” ( ๐‘ฅ ) = โˆš ๐‘ฅ + 1 , find the domain of the function ( ๐‘“ + ๐‘” ) .

  • A [ โˆ’ 1 , โˆž ) โˆ’ { 0 , 5 }
  • B ( โˆ’ โˆž , โˆ’ 1 ]
  • C โ„ โˆ’ { 0 , 5 }
  • D [ โˆ’ 1 , โˆž )
  • E [ 1 , โˆž )

Q16:

If ๐‘“ โˆถ ( โˆ’ 7 , 8 ] โ†’ โ„ ๏Šง where ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 2 ๏Šง , and ๐‘“ โˆถ [ โˆ’ 8 , 4 ] โ†’ โ„ ๏Šจ where ๐‘“ ( ๐‘ฅ ) = 4 ๐‘ฅ + 8 ๐‘ฅ + 3 ๏Šจ ๏Šจ , find ( ๐‘“ โˆ’ ๐‘“ ) ( ๐‘ฅ ) ๏Šจ ๏Šง and the domain of ( ๐‘“ โˆ’ ๐‘“ ) ๏Šจ ๏Šง .

  • A 4 ๐‘ฅ + 7 ๐‘ฅ + 5 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ 7 , 8 ]
  • B 4 ๐‘ฅ + 7 ๐‘ฅ + 5 ๏Šจ , ๐‘ฅ โˆˆ [ โˆ’ 8 , 4 ]
  • C 4 ๐‘ฅ + 7 ๐‘ฅ + 5 ๏Šจ , ๐‘ฅ โˆˆ [ โˆ’ 7 , 4 )
  • D 4 ๐‘ฅ + 7 ๐‘ฅ + 5 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ 7 , 4 ]
  • E โˆ’ 4 ๐‘ฅ โˆ’ 7 ๐‘ฅ โˆ’ 5 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ 7 , 4 ]

Q17:

If ๐‘“ โˆถ โ„ โŸถ โ„ ๏Šฐ where ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 1 7 , and ๐‘” โˆถ [ โˆ’ 2 5 , 4 ] โŸถ โ„ where ๐‘” ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 1 1 , then find ( ๐‘“ + ๐‘” ) ( ๐‘ฅ ) and its domain.

  • A ( ๐‘“ + ๐‘” ) ( ๐‘ฅ ) = 2 ๐‘ฅ โˆ’ 1 1 , ( 0 , 4 ]
  • B ( ๐‘“ + ๐‘” ) ( ๐‘ฅ ) = 2 ๐‘ฅ โˆ’ 1 7 , [ 0 , 4 ]
  • C ( ๐‘“ + ๐‘” ) ( ๐‘ฅ ) = 2 ๐‘ฅ โˆ’ 2 8 , [ 0 , 4 ]
  • D ( ๐‘“ + ๐‘” ) ( ๐‘ฅ ) = 2 ๐‘ฅ โˆ’ 2 8 , ( 0 , 4 ]

Q18:

If ๐‘“ โ„ โ†’ โ„ ๏Šง ๏Šฑ : where ๐‘“ ( ๐‘ฅ ) = 4 ๐‘ฅ + 4 ๏Šง , and ๐‘“ ( โˆ’ 9 , 6 ] โ†’ โ„ ๏Šจ : where ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 1 ๏Šจ , find and fully simplify ( ๐‘“ โˆ’ ๐‘“ ) ( ๐‘ฅ ) ๏Šง ๏Šจ and determine the domain of ( ๐‘“ โˆ’ ๐‘“ ) ๏Šง ๏Šจ .

  • A 3 ๐‘ฅ + 5 , ๐‘ฅ โˆˆ โ„ ๏Šฑ
  • B 3 ๐‘ฅ + 5 , ๐‘ฅ โˆˆ ( โˆ’ 9 , 6 ]
  • C 3 ๐‘ฅ + 5 , ๐‘ฅ โˆˆ [ โˆ’ 9 , 0 ]
  • D 3 ๐‘ฅ + 5 , ๐‘ฅ โˆˆ ( โˆ’ 9 , 0 )
  • E 3 ๐‘ฅ + 5 , ๐‘ฅ โˆˆ ( โˆ’ โˆž , 6 ]

Q19:

If ๐‘“ โ„ โ†’ โ„ ๏Šง ๏Šฑ : where ๐‘“ ( ๐‘ฅ ) = โˆ’ ๐‘ฅ โˆ’ 1 ๏Šง , and ๐‘“ ( โˆ’ 9 , 1 ) โ†’ โ„ ๏Šจ : where ๐‘“ ( ๐‘ฅ ) = 5 ๐‘ฅ โˆ’ 3 ๏Šจ , find ( ๐‘“ + ๐‘“ ) ( ๐‘ฅ ) ๏Šง ๏Šจ and the domain of ( ๐‘“ + ๐‘“ ) ๏Šง ๏Šจ .

  • A 4 ๐‘ฅ โˆ’ 4 , ๐‘ฅ โˆˆ โ„ ๏Šฑ
  • B 4 ๐‘ฅ โˆ’ 4 , ๐‘ฅ โˆˆ ( โˆ’ 9 , 1 )
  • C 4 ๐‘ฅ โˆ’ 4 , ๐‘ฅ โˆˆ [ โˆ’ 9 , 0 ]
  • D 4 ๐‘ฅ โˆ’ 4 , ๐‘ฅ โˆˆ ( โˆ’ 9 , 0 )
  • E 4 ๐‘ฅ โˆ’ 4 , ๐‘ฅ โˆˆ ( โˆ’ โˆž , 1 )

Q20:

If ๐‘“ and ๐‘” are two real functions where ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 5 ๐‘ฅ ๏Šจ and ๐‘” ( ๐‘ฅ ) = โˆš ๐‘ฅ + 4 , determine the domain of the function ( ๐‘“ โ‹… ๐‘” ) .

  • A [ โˆ’ 4 , โˆž ) โˆ’ { 0 , 5 }
  • B ( โˆ’ โˆž , โˆ’ 4 ]
  • C โ„ โˆ’ { 0 , 5 }
  • D [ โˆ’ 4 , โˆž )
  • E [ 4 , โˆž )

Q21:

Given that and find ( ๐‘“ โ‹… ๐‘“ ) ( ๐‘ฅ ) 1 2 and state its domain.

  • A 5 ๐‘ฅ โˆ’ 2 2 ๐‘ฅ + 8 2 , ๐‘ฅ โˆˆ โ„ +
  • B 5 ๐‘ฅ โˆ’ 2 2 ๐‘ฅ + 8 2 , ๐‘ฅ โˆˆ ] โˆ’ 9 , 1 ]
  • C 5 ๐‘ฅ โˆ’ 2 2 ๐‘ฅ + 8 2 , ๐‘ฅ โˆˆ [ 0 , 1 [
  • D 5 ๐‘ฅ โˆ’ 2 2 ๐‘ฅ + 8 2 , ๐‘ฅ โˆˆ ] 0 , 1 ]
  • E 5 ๐‘ฅ โˆ’ 2 2 ๐‘ฅ + 8 2 , ๐‘ฅ โˆˆ ] โˆ’ 9 , โˆž [

Q22:

Given that ๐‘“ โˆถ ( 3 , 6 ] โ†’ โ„ ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ + 1 0 ๐‘ฅ + 2 5 ๏Šง ๏Šง ๏Šจ s u c h t h a t and ๐‘“ โˆถ ( โˆ’ 1 , 9 ) โ†’ โ„ ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 3 , ๏Šจ ๏Šจ s u c h t h a t find ( ๐‘“ โ‹… ๐‘“ ) ( ๐‘ฅ ) ๏Šง ๏Šจ and state its domain.

  • A ๐‘ฅ + 7 ๐‘ฅ โˆ’ 5 ๐‘ฅ โˆ’ 7 5 ๏Šฉ ๏Šจ , ๐‘ฅ โˆˆ [ 3 , 6 )
  • B ๐‘ฅ + 7 ๐‘ฅ โˆ’ 5 ๐‘ฅ โˆ’ 7 5 ๏Šฉ ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ 1 , 9 )
  • C ๐‘ฅ โˆ’ 3 0 ๐‘ฅ + 1 0 ๐‘ฅ โˆ’ 7 5 ๏Šฉ ๏Šจ , ๐‘ฅ โˆˆ ( 3 , 6 ]
  • D ๐‘ฅ + 7 ๐‘ฅ โˆ’ 5 ๐‘ฅ โˆ’ 7 5 ๏Šฉ ๏Šจ , ๐‘ฅ โˆˆ ( 3 , 6 ]

Q23:

If ๐‘“ and ๐‘” are two real functions where ๐‘“ ( ๐‘ฅ ) = ๏ญ ๐‘ฅ + 5 0 < ๐‘ฅ < 2 , 2 ๐‘ฅ + 5 ๐‘ฅ โ‰ฅ 2 , i f i f and ๐‘” ( ๐‘ฅ ) = ๐‘ฅ , find the domain of the function ( ๐‘“ โ‹… ๐‘” ) .

  • A [ 2 , โˆž )
  • B ( 0 , 2 )
  • C ( 0 , โˆž ) โˆ’ { 2 }
  • D ( 0 , โˆž )
  • E โ„

Q24:

Given that ๐‘“ and ๐‘” are two real functions where ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ + 2 ๐‘ฅ ๏Šจ and ๐‘” ( ๐‘ฅ ) = โˆš 5 โˆ’ ๐‘ฅ , find the value of ๏€ฝ ๐‘“ ๐‘” ๏‰ ( 5 ) if possible.

  • A0
  • B โˆ’ 3 5
  • C35
  • Dundefined

Q25:

Given that ๐‘“ and ๐‘” are two real functions where ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 1 ๏Šจ and ๐‘” ( ๐‘ฅ ) = โˆš ๐‘ฅ + 5 , find the value of ๏€ฝ ๐‘” ๐‘“ ๏‰ ( โˆ’ 2 ) if possible.

  • A โˆ’ โˆš 3 3
  • Bundefined
  • C3
  • D โˆš 3 3
  • E โˆš 3

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